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- 1. Mean Variance Analysis Mean-variance analysis - used to identify optimal or efficient portfolios. We use the expected returns, variances, and covariance’s of individual investment returns Study Session 18, Reading 54
- 2. Assumptions underlying Mean Variance Analysis 1. All investors are risk averse (ie they prefer less risk to more 2. 3. 4. 5. for the same level of expected return) Expected returns for all assets are known The variance and covariance of all asset returns are known Investors only need to know the expected returns, variances, and covariance’s of returns to determine optimal portfolios. They can ignore skewness, kurtosis, and other attributes of a distribution. There are no transaction costs or taxes Study Session 18, Reading 54
- 3. Minimum Variance Frontier minimum-variance frontier - the border of a region representing all combinations of expected return and risk that are possible (the border of the feasible region). Study Session 18, Reading 54
- 4. Minimum Variance Frontier(cont.) minimum-variance portfolio - one that has the smallest variance among all portfolios with identical expected return Steps in getting minimum-variance frontier : 1. Estimation step 2. Optimization step Formula: 1. The portfolio weights sum to 100%: Study Session 18, Reading 54
- 5. The Efficient Frontier efficient frontier - the portion of the minimum-variance frontier beginning with the global minimum-variance portfolio and continuing above it Provides the maximum expected return for a given level of variance Represents all combinations of mean return and variance or standard deviation of return Investor’s portfolio selection task is greatly simplified Study Session 18, Reading 54
- 6. The Efficient Frontier(cont.) Qualities of an efficient portfolios: Minimum risk of all portfolios with the same expected return. Maximum expected return for all portfolios with the same risk. Study Session 18, Reading 54
- 7. Instability in Minimum Variance Frontier Challenges in the instability of the minimum variance : Greater uncertainty in the inputs leads to less reliability in the efficient frontier Statistical input forecasts derived from historical sample often change over time which leads to a shifting of the efficient frontier Small changes in statistical inputs can cause large changes in the historical frontier resulting in unreasonably large short positions and frequent rebalancing Study Session 18, Reading 54
- 8. Calculations related to the Mean Variance Frontier Formula: Expected return on a portfolio of two assets E(RP) = w1E(R1) + w2E(R2) Where: E(RP) - expected return on a portfolio P Wi - proportion (or weight) of the asset allocated to Asset i E(Ri) - expected return on Asset i Study Session 18, Reading 54
- 9. Calculations related to the Mean Variance Frontier (cont.) Formula: Variance of a portfolio of two assets VARp2 = w12 VAR12 + w22 VAR22 + 2w1w2 VAR1 VAR2 Where: VARp - variance of the return on the portfolio wi - proportion (or weight) of the asset allocated to Asset i VARi - variance of the return on Asset i Study Session 18, Reading 54
- 10. Calculations related to the Mean Variance Frontier (cont.) Formula: Correlation between two assets Corr1,2 = Cov1,2 /( VAR1 * VAR2) Where: Corr1,2 - correlation between two assets Cov1,2 - covariance between two assets VARi - variance of the return on Asset i Study Session 18, Reading 54
- 11. Effect of Correlation on Portfolio Diversification Diversification - to the strategy of reducing risk by combining many different types of assets When the correlation between the returns on two assets is less than +1, the potential exists for diversification benefits. As the correlation between two assets decreases, the benefits of diversification When two assets have a correlation of -1, a portfolio of the two assets exists that eliminates risk (is risk free). If the correlation between two assets declines, the efficient frontier improves. Study Session 18, Reading 54
- 12. Effect of Number of Assets on Portfolio Diversification Diversification benefits increase as the number of assets increases. Portfolio risk will fall at a decreasing rate, as the number of assets included in the portfolio rises. The standard deviation of a large, well-diversified portfolio will get closer and closer to the broad market standard deviation as the number of assets in the portfolio increases. Study Session 18, Reading 54
- 13. Equally Weighted Portfolio Risk Formula: Variance of an equally-weighted portfolio VARp2 = (1/n)* VARi2 + {(n-1)/n}* COV Where : VARp - variance of the return on the portfolio n - number of assets in the portfolio COV - average covariance of all pairings of assets in a portfolio Portfolio variance is affected by the number of assets in a portfolio and the correlation between the assets Study Session 18, Reading 54
- 14. Capital Allocation Line (CAL) capital allocation line (CAL) - describes the combinations of expected return and standard deviation of returns available to an investor from combining the optimal portfolio of risky assets with the riskfree asset Study Session 18, Reading 54
- 15. Capital Allocation Line Equation Formula: E(Rc) = Rf + (E(RT) – Rf)* STDEVc STDEVT Where: E(Rc) - expected return on an investment combination Rf - risk free rate of return E(RT) - expected return on the optimal risky portfolio STDEVc - standard deviation of the combination portfolio STDEVT - standard deviation of the optimal risky portfolio Study Session 18, Reading 54
- 16. Capital Market Line Capital Market Line (CML) - capital allocation line in a world in which all investors agree on the expected returns, standard deviations, and correlations of all portfolio risk will fall at a decreasing rate, as the number of assets included in the portfolio rises. Formula: E(Rc) = Rf + (E(RM) – Rf)* STDEVc STDEVM Where: E(Rc) - expected return on an investment combination Rf E(RM) - risk free rate of return - expected return on the market portfolio STDEVc - standard deviation of the combination portfolio STDEVM - standard deviation of the market portfolio Study Session 18, Reading 54
- 17. Capital Asset Pricing Model (CAPM) Describes the expected relationship between risk and return for individual assets. Expresses returns as a function of beta, thus simplifying risk return calculations Provides a way to calculate an asset’s expected based on its level of systematic risk, as measured by the asset’s beta. Study Session 18, Reading 54
- 18. Security Market Line (SML) Security Market Line (SML) - graph of the CAPM representing the cross-sectional relationship between the expected return for individual assets and portfolios and their systematic risk. The intercept equals the risk free rate and the slope equals the market risk premium. Study Session 18, Reading 54
- 19. Security Market Line (SML) (cont.) Security Market Line (SML) Equation: E(Ri) = RF + βi[E(RM – RF)] Where: E(Ri) - expected return on the asset RF - risk free rate of return βi - beta of the asset E(RM – RF)]- expected risk premium Study Session 18, Reading 54
- 20. CAPM equation The beta for a stock is the ratio of its standard deviation to the standard deviation of the market multiplied by its correlation with the market Study Session 18, Reading 54
- 21. CAPM equation (cont.) Market risk premium equals the expected difference in returns between the market portfolio and the risk-free asset. Study Session 18, Reading 54
- 22. Differences between the SML and CML The SML uses systematic (non diversifiable risk) as a measure of risk while the CML uses standard deviation (total risk) SML is a tool used to determine the appropriate expected (benchmark) returns for securities while the CML is a tool used to determine the appropriate asset allocation (percentages allocated to the risk-free asset and to the market portfolio) for the investor. Then SML is a graph of the capital asset pricing model while the CML is a graph of the efficient frontier. The slope of the SML represents the market risk premium while the slope of CML represents market portfolio Sharpe ratio. Study Session 18, Reading 54
- 23. The Market Model market model - regression model used to estimate betas. It assumes two types of risk: macroeconomic (systematic) or firm specific (unsystematic) risks Formula: Ri = αi + βi*RM + εi Where: Ri - return on Asset i RM - return on the market Portfolio M αi - intercept (the value of Ri when RM equals zero) βi - slope (estimate of the systematic risk for Asset i) εi - regression error with expected value equal to zero (firm-specific surprises) Study Session 18, Reading 54
- 24. Underlying Assumptions of the Market Model The expected value of the error term is zero. The errors are uncorrelated with the market return. The firm-specific surprises are uncorrelated across assets. Study Session 18, Reading 54
- 25. Market Model Predictions The expected return on Asset i depends only on the expected return on the market portfolio, E(RM), the sensitivity of the returns on Asset i to movements in the market, βi, and the average return to Asset i when the market return is zero, αi. The variance of the returns on Asset i consists of two components: a systematic component related to the asset’s beta, βi σM , and an unsystematic component related to firmspecific events. The covariance between any two stocks is calculated as the product of their betas and the variance of the market portfolio. Study Session 18, Reading 54
- 26. Application of the Market Model Simplify the calculation for estimating the covariances To trace out the minimum-variance frontier with n assets Correlation between the returns on two assets Study Session 18, Reading 54
- 27. Calculation of Adjusted and Historical Beta Historical beta is calculated by the use of the historical regression estimate derived from the market model. Often some adjustments are made to the historical beta to improve its ability to forecast the future beta. Adjusted beta is a historical beta adjusted to reflect the tendency of beta to mean revert (towards one). An adjusted beta tends to predict future beta better than historical beta does. Study Session 18, Reading 54
- 28. Multifactor Models Describe the return of an asset in terms of the risk of the asset with respect to a set of factors. Include systematic factors, which explain the average returns of a large number of risky assets. Categories: macroeconomic factor models fundamental factor models statistical factor models Study Session 18, Reading 54
- 29. Macroeconomic factor models It assume that asset returns are explained by surprises in macroeconomic risk factors The main features are systematic and priced risk factors and factor sensitivities. Investors will be compensated for bearing priced risk factors. Different assets have different factor sensitivities to the priced risk factors defined above. Study Session 18, Reading 54
- 30. Macroeconomic factor models (cont.) Formula: Return for stocks using macroeconomic model Formula: Return on portfolio using two-factor macroeconomic factor mode Study Session 18, Reading 54
- 31. Fundamental factor models It assume that asset returns are explained by multiple firm specific factors. Sensitivities are not regression slopes. Instead, the sensitivities are standardized attributes The fundamental factors are rates of return associated with each factor Study Session 18, Reading 54
- 32. Statistical factor models Applied to a set of historical returns to determine factors that explain historical returns. Two primary statistical factor models: factor analysis models - the factors are the portfolios that best explain (reproduce) historical return covariances. principal-components models - the factors are portfolios that best explain (reproduce) the historical return variances. Study Session 18, Reading 54
- 33. Arbitrage Pricing Theory (APT) An equilibrium asset-pricing k-factor model which assumes no arbitrage opportunities exist. Describes the expected return on an asset (or portfolio) as a linear function of the risk of the asset with respect to a set of factors. Makes less-strong assumptions. Study Session 18, Reading 54
- 34. Assumptions of APT Returns are derived from a multifactor model. Unsystematic risk can be completely diversified away. No arbitrage opportunities exist arbitrage opportunity - an investment opportunity that bears no risk, no cost, and yet provides a profit Study Session 18, Reading 54
- 35. APT Equation Study Session 18, Reading 54
- 36. Differences between APT and Multifactor Models Arbitrage Pricing Theory (APT) models look similar to multifactor models While APT models are equilibrium models, multifactor models are statistical regressions APT models explain the results over a single time period as functions of different factors, while multifactor models are based on data from multiple time periods Study Session 18, Reading 54
- 37. Active Risk and Return, Information Ration Active return - return in excess of the return of the benchmark Formula: Active Return = RP – RB Active risk - the standard deviation of active returns. Components: Active factor risk Active specific risk Information ratio standardizes the return achieved by a portfolio manager by dividing the return with the standard deviation of the return. Study Session 18, Reading 54
- 38. Factor and Tracking Portfolios pure factor portfolio (or simply a factor portfolio) - a portfolio that has been constructed to have a sensitivity equal to 1.0 to only one risk factor, and sensitivities of zero to the remaining factors. tracking portfolios - have a deliberately designed set of factor exposures. That is, a tracking portfolio is deliberately constructed to have the same set of factor exposures to match (“track”) a predetermined benchmark. Study Session 18, Reading 54
- 39. Implications of CAPM assumptions Two key assumptions necessary to derive the CAPM: Investors can borrow and lend at the risk-free rate. Unlimited short selling is allowed with full access to short sale proceeds. Two major implications of the CAPM: The market portfolio lies on the efficient frontier. There is a linear relationship between an asset’s expected returns and its beta. If these assumptions don’t hold, then: The market portfolio might lie below the efficient frontier. The relationship between expected return and beta might not be linear. Study Session 18, Reading 54
- 40. Impediments to Market Integration Psychological barriers Legal restrictions Transaction costs Discriminatory taxation Political risks Foreign currency risk Study Session 18, Reading 54
- 41. Factors favouring Market Integration There are many private and institutional investors who are internationally active. Many major corporations have multinational operations. Corporations and governments borrow and lend on an international scale. Study Session 18, Reading 62
- 42. Extended CAPM extended CAPM - domestic CAPM extended to the international environment is called the The risk-free rate (Rf) is the investor’s domestic risk-free rate, and the market portfolio is the market capitalizationweighted portfolio of all risky assets in the world Assumptions needed to extend CAPM Investors throughout the world have identical consumption baskets. Purchasing power parity holds exactly at any point in time. Study Session 18, Reading 62
- 43. ICAPM Equation E(r)= Rf +(βg×MrPg )+(g1×FcrP1)+(g2×FcrP2 )+...........+(gk ×FcrPk ) Where: E(r) - asset’s expected return Rrf - domestic currency risk-free rate βg - sensitivity of the asset’s domestic currency returns to changes in the global market portfolio MrPg - world market risk premium [E(rm ) - r ] E(r m) - expected return on world market portfolio g1 to gk - sensitivities of asset’s domestic currency returns to changes in the values of currencies 1 through k FcrP1 to FcrPk - foreign currency risk premiums on currencies 1 through k Study Session 18, Reading 62
- 44. Change in the Real Exchange Rate The real exchange rate is the spot exchange rate, S, multiplied by the ratio of the consumption basket price levels Formula: X = S × (PFC /PDC) The expected foreign currency appreciation or depreciation should be approximately equal to the interest rate differential Formula: E(s) = rDC - rFC, where: s - percentage change in the price of foreign currency (direct exchange rate) Study Session 18, Reading 62
- 45. Foreign Currency Risk Premium (FCRP) Foreign Currency Risk Premium (FCRP) i- s the expected exchange rate movement minus the (risk free) interest rate differential between the domestic currency and the foreign currency Study Session 18, Reading 62
- 46. Expected Return on Foreign Investments Formula: Expected return on an unhedged foreign investment E(R) =E(RFC) + E(s) Where: E(R) - Expected domestic currency return on the investment E(RFC) - Expected foreign investment return E(s) - Expected percentage currency movement Study Session 18, Reading 62
- 47. Expected Return on Foreign Investments (cont.) Formula: Expected return on an hedged foreign investment E(R) =E(RFC) + (F-S)/S Where: E(R) investment - Expected domestic currency return on the E(RFC) - Expected foreign investment return F S - Forward rate in direct quotes - Spot rate in direct quotes Study Session 18, Reading 62
- 48. Currency Exposure local currency exposure – the sensitivity of the returns in the stock denominated in the local currency to changes in the value of the local currency domestic currency exposure - because the exposure of a currency to itself is 1, domestic currency exposure is equal to local currency exposure plus 1. Study Session 18, Reading 62
- 49. Exchange Rate Exposure exchange rate exposure – the way the value of an individual company changes in response to a change in the real value of the local currency We can estimate the currency exposure of a particular firm by regressing the firm’s stock return on local currency changes. Study Session 18, Reading 62
- 50. Economic activity and exchange rate movements Two theories to explain the relationship between economic activity and exchange rate: 1. traditional model - predicts that depreciation in the value of the domestic currency will cause an increase in the competitiveness of the domestic industry and, thus, an increase in the stock value of domestic firms 2. money demand model - an increase in real economic activity leads to an increase in the demand for the domestic currency Study Session 18, Reading 62
- 51. Active portfolio management active portfolio management - refers to decisions of the portfolio manager to actively manage and monitor the broad asset allocation and security selection of the portfolio. Equilibrium is the desirable end result of active portfolio management. Study Session 18, Reading 55
- 52. Justification of active portfolio management Develop capital market forecasts for major asset classes Allocate funds across the major risky asset classes to form the optimal risky portfolio that maximizes the reward-to-risk ratio. Allocate funds between the risk-free asset and the optimal risky portfolio in order to satisfy the investor’s risk aversion. Rebalance the portfolio as capital market forecasts and investor’s risk aversion changes (also known as market timing) Study Session 18, Reading 55
- 53. Treynor Black Model Treynor-Black model - a portfolio optimization framework that combines market inefficiency and modern portfolio theory. The model is based on the premise that markets are nearly efficient. Objective: To create an optimal risky portfolio that is allocated to both a passively managed (indexed) portfolio and to an actively managed portfolio Formula: Study Session 18, Reading 55
- 54. Adjustments in Treynor Black Model Collect the time-series alpha forecasts for the analyst Calculate the correlation between the alpha forecasts and the realized alphas Square the correlation to derive the R2 Adjust (shrink) the forecast alpha by multiplying it by the analyst’s R2 Study Session 18, Reading 55
- 55. The Portfolio Management Process Important features: 1. The process is ongoing and dynamic (there are no end points, only feedback to previous steps). 2. Investments should be evaluated as to how they affect portfolio risk and return characteristics. Phases: 1. Planning 2. Execution 3. Feedback Study Session 18, Reading 56
- 56. Investment Constraints 1. Liquidity constraints - relate to expected cash outflows that will 2. 3. 4. 5. be needed at some specified time and are in excess of available income Time horizon constraints - associated with the time period(s) over which a portfolio is expected to generate returns to meet specific future needs Tax constraints - depend on how, when, and if portfolio returns of various types are taxed Legal and regulatory factors - usually associated with specifying which investment classes are not allowed or dictating any limitations placed on allocations to particular investment classes. Unique circumstances - internally generated and represent special concerns of the investor Study Session 18, Reading 56
- 57. Investment Policy Statement (IPS) investment policy statement (IPS) - a formal document that governs investment decision making, taking into account objectives and constraints. Main role of the IPS: Be readily implemented by current or future investment advisers Promote long-term discipline for portfolio decisions. Help protect against short-term shifts in strategy Study Session 18, Reading 56
- 58. Elements of the IPS A client description Identification of duties and responsibilities of parties involved. The formal statement of objectives and constraints. A calendar schedule for both portfolio performance and IPS review. Asset allocation ranges and statements regarding flexibility and rigidity when formulating or modifying the strategic asset allocation. Guidelines for portfolio adjustments and rebalancing. Study Session 18, Reading 56
- 59. Strategic Asset Allocation Strategic asset allocation is the final step in the planning stage. Common Approaches to Strategic Asset Allocation 1. Passive investment strategies – represent strategies that are not responsive to changes in expectations 2. Active investment strategies - attempt to capitalize on differences between a portfolio manager’s beliefs concerning security valuations and those in the marketplace. 3. Semi-active, risk-controlled active, or enhanced index strategies - hybrids of passive and active strategies Study Session 18, Reading 56
- 60. Factors affecting Strategic Asset Allocation 1. Risk-return 2. Capital market expectations 3. The length of the time horizon Affect of Time Horizon: The longer the investment time horizon, the more risk an investor can take on Study Session 18, Reading 56

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